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  "body": "\n## Definition\n\nLet $X$ be a [[Noetherian scheme|ND6XEY68P4]] and let $\\mathcal{L}$ be an [[invertible sheaf|7JCF2TE3V5]] on $X$.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.\nIt is *globally generated* if for every $x \\in X$ the images of $H^0(X, \\mathcal{F})$ generate the stalk $\\mathcal{F}_x$ as an $\\mathcal{O}_{X,x}$-module.\nThe sheaf $\\mathcal{L}$ is *ample* if the following holds for every [[coherent sheaf|56RXEJ8Z6S]] $\\mathcal{F}$ on $X$.\nThere is an integer $n_0$ such that $\\mathcal{F} \\otimes \\mathcal{L}^{\\otimes n}$ is globally generated for all $n \\ge n_0$.\n\n## Remarks\n\nIf $X$ is of finite type over a Noetherian ring $A$, then $\\mathcal{L}$ is ample if and only if $\\mathcal{L}^{\\otimes m}$ is very ample over $\\operatorname{Spec} A$ for some $m \\ge 1$.\nHere very ample means $\\mathcal{L}^{\\otimes m} \\cong i^*\\mathcal{O}(1)$ for an immersion $i \\colon X \\to \\mathbb{P}^N_A$.\n\nExample: on $X = \\mathbb{P}^n_k$ the [[twisting sheaf|V93Y0XYPZD]] $\\mathcal{O}(1)$ is ample.\nBy a theorem of Serre, every coherent $\\mathcal{F}$ on $\\mathbb{P}^n_k$ has $\\mathcal{F}(n)$ globally generated for $n$ large.\n\nNon-example: on $\\mathbb{P}^1_k$, $\\mathcal{L} = \\mathcal{O}(-1)$ is not ample.\nTake $\\mathcal{F} = \\mathcal{O}_X$: then $\\mathcal{L}^{\\otimes n} = \\mathcal{O}(-n)$ has no nonzero global sections for $n \\ge 1$, so it is not globally generated.\n\nAmpleness is preserved by positive tensor powers, and $\\mathcal{L}$ is ample if and only if $\\mathcal{L}^{\\otimes m}$ is ample for one $m \\ge 1$.\n",
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    "name": "Ample line bundle",
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